Fungi are basically static. But they can spread either by forming reproductive spores that are carried on wind and rain (see the page Microbial Reproduction for more details) or by growing and extending their hyphae. Remember that hyphae are chains of fungal cells.
Hyphae grow as new cells form at the tips, creating ever longer and branching chains of cells. It takes a lot to stop them, too. Hyphae are tough enough to punch through plant cell walls and the hard exoskeletons of insects (see the Fire Ant Killer story in the News section for a photo of an ant killed by a fungus).